The goal of this paper is to establish the general existence theorems for maximal elements of preference mappings by applying new fixed point theorems in p-vector spaces and locally p-convex spaces, where \(p \in (0, 1]\) . In order to do so, we first establish fixed point theorems for set-valued mappings with the local intersection property and compact upper semicontinuous (USC) set-valued mappings in p-vector, and locally p-convex spaces, respectively, by applying the functional analysis method. Our results do not only unify general existence of maximal elements for preference mappings in general equilibrium theory, but also provide some fundamental tools in supporting the study for Schauder conjecture in Hausdorff topological vector spaces. Then, applications of new fixed point theorems are also developed to give the general existence of maximal elements for preference mappings in p-vector or locally (p-) convex spaces. In addition, a new proof for the fixed point theorem of a single-valued continuous mapping in p-normed space is also given in by Appendix A for p in (0, 1] in this paper.
